Periodic solutions of singular Duffing equations under one-sided Lipschitz condition
DOI:
https://doi.org/10.12775/TMNA.2024.030Keywords
Duffing equation, Poincaré-Birkhoff twist theorem, periodic solutionAbstract
In this paper, we study the existence and multiplicity of periodic solutions for singular Duffing equations $x''+g(x)=p(t)$. When $g$ satisfies one-sided Lipschitz condition and the related time map satisfies oscillation property, we prove that the given equation has infinitely many periodic solutions.References
A. Capietto, J. Mawhin and F. Zanolin, A continuation theorem for periodic boundary value problems with oscillatory nonlinearities, Nonlinear Differ. Equ. Appl. 2 (1995), 133–163.
M. del Pino and R. Manásevich, Infinitely many T -periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations 103 (1993), 260–277.
M. del Pino, R. Manásevich and A. Montero, T -periodic solutions for some second order differential equations with singularities Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 231–243.
T. Ding and W. Ding, Resonance problem for a class of Duffing’s equations, Chin. Ann. Math. Ser. B 6 (1985), 427–432.
T. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations 105 (1993), 364–409.
T. Ding and F. Zanolin, Periodic solutions of Duffing’s equations with superquadratic potential, J. Differential Equations 97 (1992), 328–378.
C. Fabry and A. Fonda, A systematic approach to non-resonance conditions for periodically forced planar Hamiltonian systems, Ann. Mat. Pura Appl. 201 (2022), 1033–1074.
P. Habets and L. Sanchez, Periodic solutions of some Liénard equations with singularities, Proc. Amer. Math. Soc. 109 (1990), 1035–1044.
D. Hao and S. Ma, Semilinear Duffing equations crossing resonance points, J. Differential Equations 133 (1997), 98–116.
M. Krasnosel’skiı̆ and P. Zabreı̆ko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984.
A.C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 88 (1987), 109–114.
D. Qian, Time maps and Duffing equations across resonant points, Sci. China A 23 (1993), 471–479.
C. Rebelo, A note on the Poincaré–Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal. 29 (1997), 291–311.
Z. Wang, Periodic solutions of the second order differential equations with singularities, Nonlinear Anal. 58 (2004), 319–331.
J. Xia and Z. Wang, Existence and multiplicity of periodic solutions for the Duffing equation with singularity, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 625–645.
M. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl. 203 (1996), 254–269.
M. Zhang, A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1099–1114.
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Tiantian Ma, Congmin Yang, Zaihong Wang
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
